I positioned a part within my assembly and then put a fillet weld on each end of the part. Shouldn't that secure the part in that position and not allow it to roll around?
Assembly Constraints remove degrees of freedom on parts - not weld beads.
As JD mentioned, weld beads are merely for asthetics or if you want to simulate the stresses acting on the welds in the FEA simulator. The only way to constrain your part is to use Position Constraints.
Think of using Inventor like you would assemble something in the real world.
...you put the parts together with some sort of constaints (fasteners or holding fixture - these are your assembly constraints) and then weld. Once you understand assembly constraints they are actually trivially easy as it is very similar to holding the actual parts in your hands and putting them together (at least for me). I recommend that beginners use one constraint at a time and test the remaining degrees of freedom (drag the part) rather than putting on a whole bunch of constraints and then trying to figure out what they did wrong.
A lot of AutoCAD users just want to move parts to position, but the real world involves more than that (remove degrees of freedom as assembling) and it allows you to more closely digitally prototype the real world.
The people out on the shop floor have no trouble with this - yet some CAD people find it to hard?